Aprenentatge de les matemàtiques i currículum (RESUMEN EXAMEN) (2017)Resumen Inglés
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MATHS SUMMARY EXAM
TO CLASSIFY AND JUSTITFY PROBLEMS/ACTIVITIES
Curricular perspective (which contents work)
NUMBERS AND OPERATIONS: Understand numbers, ways of representing them,
relationships and number systems, understand the meanings of operations and
how they relate and make reasonable estimations.
ALGEBRA: Understand patterns and relations, represent and analyze mathematical structures using algebraic symbols, use mathematical models to represent and understand quantitative relationships.
MEASUREMENT: Understand measurable attributes of objects and the units, systems, and processes of measurement and apply appropriate techniques, tools and formulas. Ex:calculate height of a tree: with its shadow.
GEOMETRY: Analyze characteristics and properties, 2d and 3d shapes, develop mathem. arguments about geometric relationships, locations and describe spatial relationships (coordinate geometry) and use symmetry, visualization, spatial reasoning.
DATA ANALYSIS AND PROBABILITY: Collect, organize and display relevant data to answer problems, select and use appropriate statistical methods to analyze data, do predictions based on data and apply basic concepts of probability. Ex: problems with dices.
1 Structures ADDITION AND SUBSTRACTION STRUCTURES -Dylan had 3 pennies and his mother gave him 2 more pennies. How many pennies does he have in all? Q1+ C= R -Dylan had some pennies and his mother gave him 2 more pennies. Now he has 5 pennies.
How many pennies did he have to start? R-C= Q1 -Gillian had 10 shells in her bucket. She gave her brother 3 shells. How many does she have left? Q1- C= R -Gillian had some shells in her bucket. She gave 3 to her brother. Now she has 7 shells left.
How many did she start with? R- Q2= Q1 -Dylan had 3 pennies and his mother gives him some more pennies. Now he has 5 pennies.
How many pennies did his mother give him? Q1+ D= Q2 -Angela has $35 and Kelly has $42. How much more money does Kelly have? Q2- Q1= D MULTIPLICATION STRUCTURES VISUAL MODEL Repeated addition of isolated units: Barça album cards are sold in packets of 5 cards.
If we buy 4 packets, we get 20 cards.
Repeated addition of lengths: We assemble 6 towers using 4 Lego pieces in each. The new tower has 24 pieces.
Repeated movements: A kangaroo jumps 5 units in each jump. He has jumped 3 times, so he has advanced 15 units.
Area of a rectangle: A chocolate bar has 2 rows of 5 pieces. Therefore, each bar has 10 pieces.
Distribution in rows and columns: We have 7 rows of 10 stars. So, altogether there are 70 stars.
Combination / Cartesian product: We have 5 templates and 3 cardboards of 3 different colours. We can make 15 different pieces.
2 DIVISION STRUCTURES: Division “is” a repeated subtraction SHARING: How many units has each group? Or what is the magnitude that has been repeated? Given a group of objects, if we know how many parts we want to obtain, we wonder about how many objects we have to put in each group. Ex: We have 15 candies and we want to share them among 3 children. How many candies will each child get? 15 = ? candies x 3 children. Each time we give one candy to every child we’ll have 3 candies less.
PARTITION: How many groups do we have? Or how many times have we done the repetition? How many times do we repeat each magnitude/group people? When we divide we wonder about how many groups we can make, or how many times we repeat the magnitude. Given a group of objects, and knowing how many objects we want to put in each group, we wonder about how many groups we can get. Ex: We have 15 candies. If we want to give 5 candies to each child, how many children will get candies? 15 = 5 candies x ? children. Each time we give 5 candies to each child we have 5 candies less.
Productive and reproductive practice ● REPRODUCTIVE: Quick estimation activities, oral problems, addition sequences, human calculator. We must focus on their evolution! Involve context based tasks and straight-forward exercises. The focus should be on extending familiar arithmetical facts and on develop strategies to make quick calculations.
● PRODUCTIVE: Is caused by a problem solving environment. Supply the group with new problems and questions. use reproductive knowledge to solve.
Example: we have a parrot that can only say five, so we have to find as many additions or subtractions as we can that he can solve (so the result has to be five).
3 Transition between calculation by counting, calculation by structuring and formal calculation.
ADDITION AND SUBSTRACTION → From concrete objects to symbolization ● ● Calculation by counting: supported when necessary by counting materials Calculation by structuring: uses the help of suitable models to structure calculations: It allows progress from calculation by counting: • Formal calculation: using numbers as mental objects for smart and flexible calculation without the need for structured materials.
Allows going from structuring to formal calculation Strategies: Create mental images of numbers, doubling and grouping by fives and tens.
- Putting mental operations into words to realize the mental operations they have made and (foes from counting to formal…).
MULTIPLICATION Counting: children solve the problem with formal repeated additions.
Structuring: use the properties of numbers and operations for doing the multiplications (the line model (repeated addition of lengths) and the group model (repeated addition of isolated units)): 8 x 6 = 8 x 5 + 8 x 1 Formal level: No visual models. Students rely on associative and commutative properties as well as the times tables→ increasing the ability to reason.
4 1. Explain the evolution: counting-structuring-formal in the teaching and learning of operations.(addition, substraction,multiplications, divisions) (TABLE: TRANSITIONS) 2. Kinds of problems for addition, subtraction, multiplication and division.
(TABLE: STRUCTURES) 3. Multiplication tables: how to teach them and practice them.
- PROPOSAL 1: “The classic one”. The basic multiplications in different tables. Each day is assigned to one table.The method is based on repetition and practice.
- PROPOSAL 2: Looking for patterns. Ex: by noticing that in the 5 times table the numbers change by adding 5 each time: 5, 10, 15…(introduce concept of multiple). Based on repetition and practice, but the pattern help to memorize.
- PROPOSAL 3: The tables are made by the students, using strategies, such as: Commutative property, Multiplication by 2, 5, 10; Calculate doubles/halves... This proposal does focus on learning the tables using deduction and the main properties of numbers, as well as the properties of the numeral system.
4. Algorithms: Characteristics and how to teach them and practice them.
Algorithm: mechanical processes of calculation with digits.
Algorithm calculation: Is the Standard way: traditional form of calculation.
Different from Column Calculation (Reasoned): better, made with sense.
How to teach? Algorithms in: Addition and substraction→ counting-structuring-formal calculation.
1. With materials → COUNTING 2.
Operation with quantities/symbols (without materials) → STRUCTURING 5 3. The standard algorithm: opetare with units, hundreds, thousands... → FORMAL Multiplication algorithms: - Teach multiplication tables methods - Teach counting-structuring-formal calculation.
Division algorithms: The algorithm goes to left to right, and it gives two results: the quotient and the reminder. This algorithm has some restrictions: ● dividend>divisor ● dividend>reminder We have to decompose (decide which part we should take) and estimate (approximate the quocient, starting from the smallest number possible), if it does not work you try it with the successive number.
For dividing will need to apply our knowledge about substraction and multiplication algorithms.
5. Estimation: why is it important? Different types. How to teach it and why.
TYPES: Calculation with rounded off numbers: finding a global answer to a problem given with exact numbers: 78 x 81 = Calculation with estimated values: when the necessary data is incomplete or unavailable. Approximately, how many days old are you? How to teach it: phases in estimation Rounding off numbers: - Round off to the closest 10, 100, 1000… - Flexible rounding off: up and down Estimations in additions and subtractions: 543 - 178 Estimation in multiplications and divisions: The cargo box of a truck is 8.62m long, 2.09 m wide and 2.55 m high. Which is the best way to estimate the volume of the cargo space? Estimation in case of incomplete data: LOCAL NEWS: 5981 fans departed from Amsterdam in 297 buses. Could it be correct? Why/when is estimation important? When...
6 Exact calculation is not necessary: to say whether a 10 euro note is enough to pay for 4 loafs of bread of 1.98 euro each, you don’t need to calculate the price exactly.
Exact calculation is not possible: the question about how long you would have to save to buy a skateboard cannot be answered exactly because you don’t know how much will you able to save each week.
Exact calculation is not sensible: the average mileage of a car depends on so many different factors that precise calculation is not sensible.
6. Give key general ideas on teaching geometry.
Geometry in primary school has to familiarize with three-dimensional space.
Geometrical activities related to many contexts that require that children look, handle, draw, construct, describe, anticipate, justify, relate and remember facts and situations related to space and movement, emphasizing the shape and the position of objects.
The activities must be significant and must help them with their subsequent learning. Activities have to: - Be related to real situations that require the use of geometrical ideas and concepts.
- Promote visualization and representation.
- Imply the reduction of the spatial environment in different ways (analysis of the environment).
- Require the oral description of the environment from a geometrical perspective.
- Study shapes that are produced by humans and also shapes from the natural environment, using geometrical knowledge.
7. Orientation: what does it mean in primary school? What contents curriculum stablishes? What kind of activities in the classroom? Orientation in primary school To determine their own position and that of other objects in space with the help of elementary orientation terms (direction, angle, distance, parallelism, coordinates). Orientating: be able to interpret a visual model of spatial location, which is perceived from a given point of view.
Contents of the curriculum Cicle inicial - Descripció, nominació i interpretació de posicions relatives a l’espai, en referència a un mateix i a altres punts.
- Descripció i interpretació de la direcció en els desplaçaments a l’espai.
- Representació i elaboració d’itineraris senzills, laberints o plànols.
Cicle mitjà - Utilització d’adreces o punts de referència per moure’s en l’entorn proper.
- Creació i ús dels sistemes de coordenades per localitzar distàncies entre dos punts i descriure camins.
7 - Realització, interpretació i ús de plànols d’itineraris coneguts.
Cicle Superior - Representació de figures geomètriques sobre eixos de coordenades.
- Utilització d’escales sobre mapes per calcular distàncies reals.
- Localització de punts en un sistema de coordenades.
Activities We can do activities starting from a flat representation of space that is similar to a simplified map (we draw paths / sketches) that can contain iconic representations.
8. Visualization: what does it mean in primary school? What contents curriculum stablishes? What kind of activities in the classroom? Visualization in primary school To improve and develop visualization skills and help students to learn how to look and understand what they see.
Requires students to anticipate, discover and check how several objects are seen from different positions and points of view.
Contents of the curriculum - Spatial memory.
- visual-modeling activities and construction of towns, buildings… - Drawing and construction Activities - Remember and draw unknown images that they have seen only for a few seconds. The difficulty of the configuration can change depending on the year.
- Another activity would be to identify the different views from differents positions of the same figure.
9. Design a rubric assessment.
PLANIFICACIÓ Planificació anticipada: - Moment 1→ Choose a mathematical problem.
- Moment 2→ Analyse possible strategies to solve it.
Planificació de la gestió (Moment 3→ NO) Planificació curricular de l’avaluació - Moment 4.1. → Identify dimensions to assess (curr. dev).
- Moment 4.2. → Identify competences to assess (RUBRIC).
8 RUBRIC: DIMENSIONS 1) Dimensió 2) Dimensió 3) Dimensió 4) Dimensió de de de de resolució de problemes raonament i prova Connexions comunicació i representació COMPETENCES 1) Mathematic strategies to solve→ (criteris) ➢ ➢ ➢ ➢ Identify data Represent problem Explain problem Represent problem if it does not work/ change strategy.
2) Proves the solution→ (criteris) ➢ Express the solution.
➢ Treat the multiple solutions.
➢ Prove solutions.
3) Make questions and generate problems→ (criteris) ➢ Formulate problems (in context).
Criteris de resultat Criteris de realització Nivell 1 Nivell 2 Nivell 3 Nivell 4 Identifica dades amb ajuda.
Identifica les dades, però no les interpreta bé.
Identifica i interpreta les dades, però no diferencia les rellevants.
Identifica i interpreta les dades rellevants.
Representaci ó del problema Representa el problema amb ajuda.
Representa el problema mitjançant representació gràfica.
Representa el problema mitjançant esquemes, expressions aritmètiques.
Representa el problema mitjançant representació numèrica.
Replantejam ent del problema, en cas que no funcioni l’estratègia Si el problema no li surt, necessita ajudar per buscar estratègies.
Replanteja el problema des de l’inici si una estratègia no li funciona.
Refà tot el procés si l’estratègia no li funciona.
Torna enrere, detecta el punt conflictiu i el refà.
Explica el problema amb exemples gràfics.
Verbalitza l’estratègia usada i explica el procés seguit.
Justifica el procés usant llenguatge matemàtic.
Identificació de dades del problema Explicació del Explica el procés problema amb ajuda.
9 10.Propose a sequence of actions (“teaching unit”) for teaching decimal system, addition, subtraction, multiplication, division, shape, orientation and position, and visualization.
Decimal system: Multibase Blocks help us to teach it: Addition: Substraction: Multiplication: Division: Shape: Orientation: Position: Visualization: 10 ...